Shapes of RNA pseudoknot structures

TL;DR

This paper analyzes the abstract shapes of complex RNA pseudoknot structures, deriving generating functions and asymptotic formulas to understand their combinatorial properties.

Contribution

It introduces a novel approach to compute generating functions for generalized RNA shape classes and derives explicit asymptotic expressions for their enumeration.

Findings

Derived generating functions for RNA shape classes

Obtained explicit asymptotic formulas for shape enumeration

Extended shape analysis to complex RNA pseudoknot structures

Abstract

In this paper we study abstract shapes of $k$-noncrossing, $\sigma$-canonical RNA pseudoknot structures. We consider ${\sf lv}_k^{\sf 1}$- and ${\sf lv}_k^{\sf 5}$-shapes, which represent a generalization of the abstract $\pi'$- and $\pi$-shapes of RNA secondary structures introduced by \citet{Giegerich:04ashape}. Using a novel approach we compute the generating functions of ${\sf lv}_k^{\sf 1}$- and ${\sf lv}_k^{\sf 5}$-shapes as well as the generating functions of all ${\sf lv}_k^{\sf 1}$- and ${\sf lv}_k^{\sf 5}$-shapes induced by all $k$-noncrossing, $\sigma$-canonical RNA structures for fixed $n$. By means of singularity analysis of the generating functions, we derive explicit asymptotic expressions.